This depends on whether you are still in algebra or in calculus.
I'll do it both ways:

1.  Using calculus:

By implicit differentiation:





We set  equal to 0





, substituting:













Answer, since the + gives the larger value, 
the maximum point is 

-------------------------------

2.  If you're in algebra,



Swap the middle two terms to get like lettered terms together:



Add 3 to both sides:



Factor 4 out of the first two terms:



Complete the square inside the parentheses:
                
                
                
Add +1 inside the parentheses and add +4*1 to the right side
since 4 times what is added inside the parentheses is added
to the left side and must also be added to the right side.



Complete the square on the y's:
                
                

Add +1 to both sides:



Factor the two trinomials, which are perfect squares, and
write them as the squares of binomials, combine numers
on the right side:



Divide through by 8





Compare to the standard equation for an ellipse:

 since a > b

center (h,k) = (-1,1), , 

Graph:



The highest point on the ellipse is the upper vertex, which has the
same x-coordinate as the center, which is -1,

The y-coordinate of the upper vertex, which is  units 
above the center, which has y-coordinate 1, so we add  to
1 and get .

Thus the highest point on the ellipse is:



Edwin